The demand for electricity varies greatly during the day. Because large amounts of electricity cannot be stored economically, electric power companies cannot manufacture electricity and hold it in inventory until it is needed. Instead, power companies must balance the production of power with the demand for power in real time. One of the greatest uncertainties in forecasting the demand for electricity is the weather. Most power companies employ meteorologists who constantly monitor weather patterns and update computer models that predict the demand for power over a rolling, seven-day planning horizon. This forecasted seven-day window of demand is referred to as the company’s load-profile and is typically updated every hour.

Every power company has a base-load demand that is relatively constant. To satisfy this base-load demand, a power company uses its most economical, low-cost power-generating assets and keeps them running continuously. To meet additional demands for power above the base-load, a power company must dispatch (or turn on) other generators. These other generators are sometimes called “peakers” because they help the power company meet the highest demands or peak loads. It costs different amounts of money to bring different types of peakers online. And because different peakers use different types of fuel (for example, coal, gas, bio-mass), their operating costs per megawatt (MW) generated also differ. Thus, dispatchers for a power company continually have to decide which generator to bring online or turn off to meet their load profile in the least costly manner.

The Old Dominion Power (ODP) Company provides electrical power throughout Virginia and the Carolinas. Suppose ODP’s peak-load profile (that is, the estimated load above base-load) in megawatts (MWs) is currently estimated as follows:

   

Day
	1	2	3	4	5	6	7
Load (in MWs)	4,300	3,700	3,900	4,000	4,700	4,800	3,600

ODP currently has three peaking generators offline that are available to help meet this load. The generators have the following operating characteristics:

    

Generator Location	Startup Cost	Cost per Day	Maximum MW Capacity per Day
New River	$    800	$200 + $5 per MW	2,100
Galax	$1,000	$300 + $4 per MW	1,900
James River	$    700	$250 + $7 per MW	3,000

To get an offline generator up and running, a startup cost must be paid. After a generator is running, it can continue to run indefinitely without having to pay this startup cost again. However, if the generator is turned off at any point, the setup cost must be paid again to get it back up and running. Each day that a generator runs, there is both a fixed and variable cost that must be paid. For example, any day that the New River generator is online, it incurs a fixed cost of $200 plus $5 per MW generated. So even if this generator is not producing any MWs, it still costs $200 per day to keep it running (to avoid a restart). When they are running, each generator can supply up to the maximum daily megawatts listed in the final column of the table.

1. Formulate a mathematical programming model for ODP’s power-dispatching problem.

2. Implement your model in a spreadsheet and solve it.

3. What is the optimal solution?

4. Suppose ODP can sometimes buy power from a competitor. How much should ODP be willing to pay to acquire 300 MW of power on day 1? Explain your answer.

5. What concerns, if any, would you have about implementing this plan?